Research Article Structural Parameter Identification of...

Research ArticleStructural Parameter Identification of Articulated ArmCoordinate Measuring Machines

Guanbin Gao, Huaishan Zhang, Xing Wu, and Yu Guo

Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming, China

Correspondence should be addressed to Guanbin Gao; [email protected]

Received 29 February 2016; Revised 14 June 2016; Accepted 4 September 2016

Academic Editor: Tomonari Furukawa

Copyright © 2016 Guanbin Gao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Precise structural parameter identification of a robotic articulated arm coordinate measuring machine (AACMM) is essential forimproving its measuring accuracy, particularly in robotic applications. This paper presents a constructive parameter identificationapproach for robotic AACMMs. We first develop a mathematical kinematic model of the AACMM based on the Denavit-Hartenberg (DH) approach established for robotic systems. This model is further calibrated and verified via the practical test data.Based on the difference between the calculated coordinates of the AACMMprobe via the kinematic model and the given referencecoordinates, a parameter identification approach is proposed to estimate the structural parameters in terms of the test data set.TheJacobian matrix is further analyzed to determine the solvability of the identification model. It shows that there are two couplingparameters, which can be removed in the regressor. Finally, a parameter identification algorithm taking the least-square solutionof the identification model as the structural parameters by using the obtained poses data is suggested. Practical experiments basedon a robotic AACMM test rig are carried out, and the results reveal the effectiveness and robustness of the proposed identificationapproach.

1. Introduction

The coordinate measuring machine (CMM) is a universalmeasuring instrument which can transform various geomet-ric measurements into coordinate measurements [1]. Thisinstrument has been widely used in the calibration andmodeling of robotic systems. In particular, the articulatedarm CMM (AACMM) is a new type of robot-like CMMwithmultiple degrees of freedom (DOF), which generally consistsof a series of linkages connected by joints in series [2, 3]. TheAACMM obtains the angles of joints by means of the angleencoder installed on the rotary joints. And the angles canbe transformed into the three coordinates (𝑥, 𝑦, 𝑧) throughthe kinematic model. The AACMM possesses some specificand essential characteristics and advantages, for example,simple mechanical structure, small size, light weight, largemeasurement range, and flexible measurement in field [4].

However, the measuring accuracy of the AACMM ismuch lower than that of the orthogonal CMM [5, 6], whichmay greatly limit its applications. A potential strategy toimprove the measuring accuracy of the AACMM is to choose

high precision hardware components and to apply highrequirements for manufacturing and assembly [3, 7]. How-ever, the cost of the measuring machine increases dramati-cally by using more precise components. Moreover, in somespecific applications, increasing the precision of individualcomponents may not be able to ensure overall increasedmeasuring precision and accuracy. In particular, small errorsin some kinematic parameters may accumulate and thusinfluence the measuring accuracy of the AACMM greatly.

Another essential and economic way to eliminate errorsof the structural parameters and to improve the measuringaccuracy is to identify the robot’s structural parameters [8] byusing appropriate parameter identification approaches. Gen-erally, the structural parameter identification includes foursteps [9]: (1) modeling: to establish a mathematical modeldescribing the geometrical characteristics and kinematics ofthe robot; (2) measurement: to measure the coordinates ofthe end effector in the real world coordinate system; (3)identification: to identify the structural parameters of theobtained model by means of mathematical calculation ofthe data set; (4) compensation: to modify the parameters

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 4063046, 10 pageshttp://dx.doi.org/10.1155/2016/4063046

2 Mathematical Problems in Engineering

in the control system according to the identification results.These four steps are also applicable for the AACMM, and animproved identification method will be studied in this paper.

For the parameter identification of AACMM, Kovac andFrank [10] developed a new high precision device for theAACMMtesting and calibrationwith the laser interferometermeasurements along a line gauge beam. Santolaria et al.[11, 12] reported a method to calibrate an AACMM based onthe Denavit-Hartenberg (DH) kinematic model parameters.These parameters are optimized by measuring a calibratedball bar gauge located at different orientations and positionsin the AACMM working space. Hamana et al. [13] presenteda method, where the kinematic parameters of AACMMwerecalibrated using spherical center coordinates as the artifact.However, only part of the measuring space can be cali-brated with the above methods. Thus, the AACMM cannotbe calibrated by directly using these available results. Inparticular, the coupling relationships between the structuralparameters and their effect on the measuring uncertainty ofthe AACMM were not considered in the above researches.Therefore, the robustness and efficiency of the identificationswere affected due to those invalid calculations [14] of theredundant couplings.

In this paper, we propose an improved modeling andparameter identification method for AACMM robotic sys-tem. First, the kinematic model and structural identificationmatrix were established based on the DH method, andthe coupling relationship between the structural parameterswas obtained through further analysis of the structuralidentification matrix. Then the identification model of theAACMM was constructed, and a parameter estimationapproach developed based on the LS method is proposedto identify the structural parameters. Practically collecteddata of the joint angles and coordinates of the probe areused to validate the model and identification approach. Theredundancy embedded in the parameter matrix is furtheranalyzed and eliminated to address the coupling effects andidentifiability. Finally, practical experiments are conducted toverify the efficiency of the proposed identification method.

The advantages and the distinctive features of this pro-posed identification method in comparison to some otheridentification methods for AACMM (e.g., [12, 15, 16]) are asfollows:

(1) We do not need precise initial parameters, and evenwe do not need initial parameters (we can assign theinitial parameters arbitrarily as long as they are nottoo exaggerated) in the identification. The identifiedvalues of the structural parameters can be solvedthrough (18). However, in some available results, forexample, [12, 15, 17], the initial identification parame-ters should be appropriately selected to achieve goodidentification results because an iteration calculationmethod is adopted.

(2) In this paper, we conducted the coupling analysissuch that those linearly dependant parameters aredetected and removed from the parameters to beidentified. Consequently, the calculation costs and theidentification efficacy can be significantly improved.

Base

Joint 1

Joint 2

Joint 4

Joint 5

Joint 6

Probe

Joint 3

Figure 1: The structure of the AACMM.

In fact, in our case study, only one time iterationcalculation can provide fairly good results.

(3) The time consumed by the proposed identificationcalculation is relatively shorter than the widely usediteration identificationmethods such as PSO,GA, andLS; that is, we can get the results just after one timeiteration calculation.

The paper is organized as follows. Section 2 presents thekinematic modeling and validation; the parameter identi-fication and the analysis are introduced in Section 3; andexperimental results are given in Section 4. Section 5 pro-vides conclusions.

2. Kinematic Modeling and Verification

2.1. Kinematic Modeling. As shown in Figure 1, the structureof the AACMM is similar to an articulated robot. Therefore,the AACMM model can be established by using availablemodeling methods developed for robotics. For the modelingof robotic kinematics, the most influential method is theDenavit-Hartenberg model (DH model) which has beenwidely used due to its clear physical meaning [18]. A 4 ×4 homogeneous transform matrix is used to represent thespatial relations of adjacent joints coordinate systems [19].Because all the adjacent joints of the AACMM are perpen-dicular [20], there is no nominally parallel problem, and wecan use the DH method to establish the kinematic model.

In this paper, a 6-DOFAACMMis studied.The schematicstructure of this system is shown in Figure 1. For this 6-DOFAACMM, the coordinate values of the probe in the referencecoordinate system can be derived through 7 successivetransformations.

Mathematical Problems in Engineering 3

O1O2

O3

d1

d3

d4

O5

O6O4

d5

l

X1X2

X3

X4

X5X6

Y3

Y4

Y5

Y6

X0

Y2

Y0

Y1

Z6

Z5Z4

Z3

Z2

Z1

Z0

O0

Figure 2: Coordinate systems of the AACMM.

Table 1: The estimated structural parameters of the AACMM.

Linkage number 𝑖 𝑎𝑖 [mm] 𝑑𝑖 [mm] Δ𝜃𝑖 [∘] 𝛼𝑖 [∘]1 0 376 0 −902 62 0 0 −903 0 751 0 −904 62 0 0 −905 0 500 0 −906 0 15 0 90𝑙 = 98mm.

According to the DH method, there are four groups ofstructural parameters in the AACMM, for example, linkagelength 𝑑𝑖, joint length 𝑎𝑖, torsion angle 𝛼𝑖, and joint angle𝜃𝑖.

The coordinates of the AACMM were established fol-lowed by the DH method, as shown in Figure 2. Someof the four groups of structural parameters were achievedvia direct measurement and some of them can only bedetermined by using estimation method because they cannotbemeasured directly, for example, 𝛼𝑖, as shown in Table 1.Themeasurement datum of the structural parameters is all axes ofthe joints and linkages, which are virtual lines only, and thepositions of them are not accurate. Therefore, the structuralparameters are not accurate as they will result in significantmovement or measurement uncertainty of the AACMM.

According to the principle of homogeneous transforma-tion, the transform from the coordinate system {𝑂𝑖𝑋𝑖𝑌𝑖𝑍𝑖}to {𝑂𝑖−1𝑋𝑖−1𝑌𝑖−1𝑍𝑖−1} is equivalent to a process in which{𝑂𝑖−1𝑋𝑖−1𝑌𝑖−1𝑍𝑖−1} conducts rotation and translation and

Figure 3:The interface of the acquisition software of the AACMM.

then completely coincides with {𝑂𝑖𝑋𝑖𝑌𝑖𝑍𝑖}. The whole trans-form procedure can be expressed mathematically as

𝑇𝑖−1,𝑖 = Rot (𝑧𝑖−1, 𝜃𝑖)Trans (0, 0, 𝑑𝑖)Trans (𝑎𝑖, 0, 0)⋅ Rot (𝑥𝑖, 𝛼𝑖)

= [[[[[[

cos 𝜃𝑖 − sin 𝜃𝑖 cos𝛼𝑖 sin 𝜃𝑖 sin 𝛼𝑖 𝑎𝑖 cos 𝜃𝑖sin 𝜃𝑖 cos 𝜃𝑖 cos 𝛼𝑖 − cos 𝜃𝑖 sin 𝛼𝑖 𝑎𝑖 sin 𝜃𝑖0 sin 𝛼𝑖 cos 𝛼𝑖 𝑑𝑖0 0 0 1

]]]]]].

(1)

The probe coordinates in the reference coordinate system{𝑂0𝑋0𝑌0𝑍0} can be expressed as follows:

[[[[[[

𝑥𝑦𝑧1

]]]]]]= 𝑇0,1 ⋅ 𝑇1,2 ⋅ 𝑇2,3 ⋅ 𝑇3,4 ⋅ 𝑇4,5 ⋅ 𝑇5,6 ⋅

[[[[[[

00𝑙1

]]]]]]

= 6∏𝑖=1

[[[[[[

cos 𝜃𝑖 − sin 𝜃𝑖 cos𝛼𝑖 sin 𝜃𝑖 sin 𝛼𝑖 𝑎𝑖 cos 𝜃𝑖sin 𝜃𝑖 cos 𝜃𝑖 cos 𝛼𝑖 − cos 𝜃𝑖 sin 𝛼𝑖 𝑎𝑖 sin 𝜃𝑖0 sin 𝛼𝑖 cos 𝛼𝑖 𝑑𝑖0 0 0 1

]]]]]]

⋅ [[[[[[

00𝑙1

]]]]]].

(2)

There is one group of parameters which are the variablesin (2). It is the joint angle 𝜃𝑖, the value of which can beacquired by the joint angular sensor.

2.2. Verification of the Kinematic Model. To validate theproposed kinematic model (2), we collect realistic operationdata via sensors, for example, joint angle of the AACMM,which are then used to calculate the coordinate system. Forthis purpose, a software according to the kinematic model(2) was developed to acquire the joint angles, which can alsobe used to verify the kinematic model of the AACMM. Theinterface of the built software with C++ is shown in Figure 3.As we find all the structural parameters can be imported intothe software through the interface. Then with the joint angles


acquired from the AACMM and the structural parameters,the coordinates of the AACMM probe can be calculated bythe software.

By comparing the coordinates of the probe calculated bythe software and the reference coordinates (which can betreated as the true values of the coordinates of the probe), wefound that the calculated coordinates of the probe are veryclose to the reference coordinates. The differences betweenthe two group coordinates are always less than 1.5mm, whichmeans that the movement uncertainty of the AACMM isabout 1.5mm. In this sense, the kinematic model (2) of theAACMM is correct because a wrong kinematic model willlead to uncontrollable movement uncertainty. However, itshould be noted that although such movement uncertaintyis acceptable for industrial robots, it is not applicable forAACMM as this is a much more precision instrument. Thusfurther analysis will be conducted in the following Section 3to identify the exact structural parameters by introducing adiscrete-time parameter identification method.

3. Identification for AACMMUncertain Parameters

3.1. Modeling Error Analysis. It is known that the accuracyof the AACMM is affected greatly by the manufacture,assembly, and component selection [21, 22], which can allresult in inconsistence between the structural parametersin the kinematic model and the actual ones. However, theproposed identification method in this paper can achievehigh accuracy for AACMM modeling without increasingthe cost to construct the parts and buy extra sensors. Thissection will further address the improvement of accuracyby analyzing the error dynamics and then introducing anassociated identification scheme.

From (2), we know that the uncertainty of the AACMMis affected by the error of structural parameters includingthe linkage length error Δ𝑑𝑖, the torsion angle error Δ𝛼𝑖, the

joint length error Δ𝑎𝑖, the probe length error Δ𝑙, and thejoint angle deviation Δ𝜃𝑖 at the zero position [23]. Thus theeffects of these uncertainties should be identified and thencompensated. For this purpose, (2) can be rewritten as afunction group of the structural parameters, which is shownin

𝑥 = 𝑓𝑥 (𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝛼1, 𝛼2, 𝛼3, 𝛼4, 𝛼5, 𝛼6, 𝑑1, 𝑑2, 𝑑3, 𝑑4,𝑑5, 𝑑6, 𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6, 𝑙)

𝑦 = 𝑓𝑥 (𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝛼1, 𝛼2, 𝛼3, 𝛼4, 𝛼5, 𝛼6, 𝑑1, 𝑑2, 𝑑3, 𝑑4,𝑑5, 𝑑6, 𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6, 𝑙)

𝑧 = 𝑓𝑥 (𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝜃1, 𝛼1, 𝛼2, 𝛼3, 𝛼4, 𝛼5, 𝛼6, 𝑑1, 𝑑2, 𝑑3, 𝑑4,𝑑5, 𝑑6, 𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6, 𝑙) .

(3)

To show the effect of uncertainties, we calculate thedifferential of (3) as

𝑑𝑥 = 6∑𝑖=1

𝜕𝑓𝑥𝜕𝜃𝑖 𝑑𝜃𝑖 +6∑𝑖=1

𝜕𝑓𝑥𝜕𝛼𝑖 𝑑𝛼𝑖 +6∑𝑖=1

𝜕𝑓𝑥𝜕𝑑𝑖 𝑑𝑑𝑖 +6∑𝑖=1

𝜕𝑓𝑥𝜕𝑎𝑖 𝑑𝑎𝑖+ 𝜕𝑓𝑥𝜕𝑙 𝑑𝑙

𝑑𝑦 = 6∑𝑖=1

𝜕𝑓𝑦𝜕𝜃𝑖 𝑑𝜃𝑖 +6∑𝑖=1

𝜕𝑓𝑦𝜕𝛼𝑖 𝑑𝛼𝑖 +6∑𝑖=1

𝜕𝑓𝑦𝜕𝑑𝑖 𝑑𝑑𝑖 +6∑𝑖=1

𝜕𝑓𝑦𝜕𝑎𝑖 𝑑𝑎𝑖+ 𝜕𝑓𝑦𝜕𝑙 𝑑𝑙

𝑑𝑧 = 6∑𝑖=1

𝜕𝑓𝑧𝜕𝜃𝑖 𝑑𝜃𝑖 +6∑𝑖=1

𝜕𝑓𝑧𝜕𝛼𝑖 𝑑𝛼𝑖 +6∑𝑖=1

𝜕𝑓𝑧𝜕𝑑𝑖𝑑𝑑𝑖 +6∑𝑖=1

𝜕𝑓𝑧𝜕𝑎𝑖 𝑑𝑎𝑖+ 𝜕𝑓𝑧𝜕𝑙 𝑑𝑙.

(4)

Thus (4) can be rewritten in a compact form as given in

[[[𝑑𝑥𝑑𝑦𝑑𝑧

]]]=[[[[[[[[[[

𝜕𝑓𝑥𝜕𝜃1 ⋅ ⋅ ⋅𝜕𝑓𝑥𝜕𝜃6

𝜕𝑓𝑥𝜕𝛼1 ⋅ ⋅ ⋅𝜕𝑓𝑥𝜕𝛼6

𝜕𝑓𝑥𝜕𝑑1 ⋅ ⋅ ⋅𝜕𝑓𝑥𝜕𝑑6

𝜕𝑓𝑥𝜕𝑎1 ⋅ ⋅ ⋅𝜕𝑓𝑥𝜕𝑎6

𝜕𝑓𝑥𝜕𝑙𝜕𝑓𝑦𝜕𝜃1 ⋅ ⋅ ⋅

𝜕𝑓𝑦𝜕𝜃6𝜕𝑓𝑦𝜕𝛼1 ⋅ ⋅ ⋅

𝜕𝑓𝑦𝜕𝛼6𝜕𝑓𝑦𝜕𝑑1 ⋅ ⋅ ⋅

𝜕𝑓𝑦𝜕𝑑6𝜕𝑓𝑦𝜕𝑎1 ⋅ ⋅ ⋅

𝜕𝑓𝑦𝜕𝑎6𝜕𝑓𝑦𝜕𝑙

𝜕𝑓𝑧𝜕𝜃1 ⋅ ⋅ ⋅𝜕𝑓𝑧𝜕𝜃6

𝜕𝑓𝑧𝜕𝛼1 ⋅ ⋅ ⋅𝜕𝑓𝑧𝜕𝛼6

𝜕𝑓𝑧𝜕𝑑1 ⋅ ⋅ ⋅𝜕𝑓𝑧𝜕𝑑6

𝜕𝑓𝑧𝜕𝑎1 ⋅ ⋅ ⋅𝜕𝑓𝑧𝜕𝑎6

𝜕𝑓𝑧𝜕𝑙

]]]]]]]]]]

× [𝑑𝜃1 ⋅ ⋅ ⋅ 𝑑𝜃6 𝑑𝛼1 ⋅ ⋅ ⋅ 𝑑𝛼6 𝑑𝑑1 ⋅ ⋅ ⋅ 𝑑𝑑6 𝑑𝑎1 ⋅ ⋅ ⋅ 𝑑𝑎6 𝑑𝑙]𝑇 .

(5)

It is shown that (5) is in a linear form with perturbingmodeling uncertainties. Since we consider sufficiently smallsampling interval in the discrete-time implementation, the

structural parameters of the AACMM are relatively small;that is, Δ𝜃𝑖 = 𝑑𝜃𝑖 and 𝑑𝑥 = Δ𝑥. Then (5) can be rewrittenas

[[Δ𝑥Δ𝑦Δ𝑧

]]= J × [Δ𝜃1 ⋅ ⋅ ⋅ Δ𝜃6 Δ𝛼1 ⋅ ⋅ ⋅ Δ𝛼6 ⋅ ⋅ ⋅ Δ𝑑1 ⋅ ⋅ ⋅ Δ𝑑6 ⋅ ⋅ ⋅ Δ𝑎1 ⋅ ⋅ ⋅ Δ𝑎6 𝑑𝑙]𝑇 . (6)


We denote

J

=[[[[[[[[[[

𝜕𝑓𝑥𝜕𝜃1 ⋅ ⋅ ⋅𝜕𝑓𝑥𝜕𝜃6

𝜕𝑓𝑥𝜕𝛼1 ⋅ ⋅ ⋅𝜕𝑓𝑥𝜕𝛼6

𝜕𝑓𝑥𝜕𝑑1 ⋅ ⋅ ⋅𝜕𝑓𝑥𝜕𝑑6

𝜕𝑓𝑥𝜕𝑎1 ⋅ ⋅ ⋅𝜕𝑓𝑥𝜕𝑎6

𝜕𝑓𝑥𝜕𝑙𝜕𝑓𝑦𝜕𝜃1 ⋅ ⋅ ⋅

𝜕𝑓𝑦𝜕𝜃6𝜕𝑓𝑦𝜕𝛼1 ⋅ ⋅ ⋅

𝜕𝑓𝑦𝜕𝛼6𝜕𝑓𝑦𝜕𝑑1 ⋅ ⋅ ⋅

𝜕𝑓𝑦𝜕𝑑6𝜕𝑓𝑦𝜕𝑎1 ⋅ ⋅ ⋅

𝜕𝑓𝑦𝜕𝑎6𝜕𝑓𝑦𝜕𝑙

𝜕𝑓𝑧𝜕𝜃1 ⋅ ⋅ ⋅𝜕𝑓𝑧𝜕𝜃6

𝜕𝑓𝑧𝜕𝛼1 ⋅ ⋅ ⋅𝜕𝑓𝑧𝜕𝛼6

𝜕𝑓𝑧𝜕𝑑1 ⋅ ⋅ ⋅𝜕𝑓𝑧𝜕𝑑6

𝜕𝑓𝑧𝜕𝑎1 ⋅ ⋅ ⋅𝜕𝑓𝑧𝜕𝑎6

𝜕𝑓𝑧𝜕𝑙

]]]]]]]]]]

(7)

as the Jacobian matrix,

ΔP = [[[Δ𝑥Δ𝑦Δ𝑧

]]]

(8)

as the difference between the coordinates of the probe cal-culated by the kinematic model and the measured referencecoordinates, and

ΔX = [Δ𝜃1 ⋅ ⋅ ⋅ Δ𝜃6 Δ𝛼1 ⋅ ⋅ ⋅ Δ𝛼6 Δ𝑑1 ⋅ ⋅ ⋅ Δ𝑑6 ⋅ ⋅ ⋅ Δ𝑎1 ⋅ ⋅ ⋅ Δ𝑎6 𝑑𝑙]𝑇 (9)

as the uncertainties perturbing the structure parameters.Then (6) can also be simplified as

ΔP = J ⋅ ΔX, (10)

where ΔX = X(𝑘) − X(𝑘 − 1) denotes the difference of theregressor within each iteration step [𝑘 − 1, 𝑘].

Equation (10) is in a strictly linearly parameterized formwithin each iteration step, and thus it can be used for thepurpose of structure parameter identification to estimate ΔX.3.2. Identification of Structural Parameters. From (6), weknow that (10) can be used to solve the error of structuralparametersΔ𝑋. It should be noted that there are 25 unknownparameters in (10), for example, Δ𝜃𝑖, Δ𝛼𝑖, Δ𝑑𝑖, Δ𝑎𝑖, 𝑖 =1 ⋅ ⋅ ⋅ 6, and Δ𝑙. However, there are only 3 measurements (e.g.,Δ𝑥, Δ𝑦, Δ𝑧). Therefore, we need to collect at least 9 groupsof reference coordinates and poses (joint angles) in variousoperation regimes to creates 9 × 3 = 27 equations, whichare then used to identify these 25 unknown parameters. Toget more robust results, we need even more than 9 groups ofreference coordinates and joint angles. Thus, for 𝑛 ≥ 9, wecan write augmented (10) as

[[[[[[[

ΔP1ΔP2...ΔP𝑛

]]]]]]]=[[[[[[[

J1J2...J𝑛

]]]]]]]⋅ ΔX, 𝑛 > 9. (11)

It is clear that the uncertain structural parameters ΔXcan be calculated based on (11) when the regressor or matrixis nonsingular. This is the well-known persistent excitationcondition and can be fulfilled in our case study by operatingthe robotic in extensive scenarios. To solve ΔX in (11)conveniently, (11) can be further rewritten in a more compactform

ΔX = (J𝑇 ⋅ J)−1 JΔP, (12)

where ΔP = [ΔP1, ΔP2, . . . , ΔP𝑛]𝑇 and J = [J1, J2, . . . , J𝑛]𝑇denote the augmented regressor and measurement matrices.

3.3. Analysis of Couplings of the Structural Parameters. In (12),if the Jacobian matrix J is not a column full rank matrix,that is, there are linear correlation structural parameters, wecannot calculate its solution directly. Therefore, we shouldcheck if there are linearly correlated structural parameters inJ. Based on the matrix theory, the linearly related row can befound by applying singular value decomposition elementaryrow transformation of J𝑇 ⋅ J via the orthogonal matrixdecomposition [24, 25]. Premultiplying J𝑇 on both sides of(12), we can obtain

[J𝑇J] ΔX = J𝑇ΔP. (13)

We denote H = [J𝑇J]; then by using the matrix singularvalue decomposition, we have

H = U[S 00 0]V𝑇, (14)

where U and V are orthogonal matrices of 25 × 25, S =diag(𝜎1, 𝜎2, . . . , 𝜎𝑟) (𝑟 ≤ 25) and r is the rank of the matrix Hand the Jacobian matrix J. Therefore, the number of linearlyrelated parameters is 25−𝑟. From (13) and (14), we can obtain

V𝑇ΔX = [S−1 00 0]U−1J𝑇ΔP. (15)

Since H is a symmetric matrix, V𝑇 = U−1, and V arerotation matrices, then V𝑇 ⋅ ΔX is equivalent to ΔX. In thiscase, the linearly related structural parameters in ΔX can befound by the elementary row transformation of the last 4lines, as shown in

ΔP = J1ΔX1, (16)

where J1 is a matrix of (3𝑛)×𝑟,ΔX1 is a vector of 𝑟 × 1, ΔP isthe matrix of (3𝑛) × 1, and 𝑟 is the rank of J1. The calculationresult shows that the rank of J1 is 23; that is, 𝑟 = 23; thus thereare two linearly related structural parameters. Specifically,we can obtain linearly related parameters by carrying out


Initial structural parameters 𝐗r

k = 0

Calculate the coordinates and 𝐉r,kPoses

Reference coordinates 𝐏t

k = k + 1

𝐗r,k = 𝐗r,k−1 + Δ𝐗r,k

Display the results

Terminal

SolveΔ𝐗r,k = (𝐉r,kT· 𝐉r,k)

−1𝐉r,k

Δ𝐏k = 𝐏t − 𝐏k

Pk

ΔPk

Figure 4: The flowchart of the identification procedure.

the singular value decomposition and the elementary rowtransform. As shown in (17), we find that Δ𝑎6 is linearlyrelated with Δ𝜃6 and Δ𝑑6 is linearly related with Δ𝛼6. Thus inthe identification calculation, Δ𝜃6 and Δ𝛼6 are treated as theredundant parameters and they do not need to be identified.Also, Δ𝑎6 and Δ𝑑6 can be determined as the redundantparameters and are not identified in the identification calcu-lation, which can also reduce the dimension of the regressormatrices and thus the required calculation costs.

Δ𝑎6 = 𝑙Δ𝜃6Δ𝑑6 = 𝑙Δ𝛼6. (17)

3.4. Implementation of Identification Calculation. Accordingto the analysis presented in last section, we know that thereare two structural parameters which are related to others anddo not need to be identified in terms of (12). Instead, they canbe calculated based on (17) provided other 23 parameters areidentified. Therefore, the two corresponding columns of theJacobianmatrix J1 can be removed and the new identificationequations can be obtained as shown in

ΔX𝑟 = (J𝑟𝑇 ⋅ J𝑟)−1 J𝑟ΔP, (18)

where ΔX𝑟 = X𝑟(𝑘) − X𝑟(𝑘 − 1) denotes the change betweeneach iteration interval [𝑘 − 1, 𝑘], which is the vector of thestructural parameters of the AACMM after the removal of 𝑎6and 𝑑6, and J𝑟 is the Jacobian matrix after removal of the twocolumns of 𝑎6 and 𝑑6.

The overall identification algorithm has been carried outto solve (18) in Matlab, and the flowchart shown in Figure 4provides the practical implementation of the suggested algo-rithm.

In the literature, there are many different identificationmethods, for example, PSO and GA. However, these algo-rithms need much more computational costs, which maylimit their applicability for AACMM. The time consumed bythe proposed identification calculation is relatively shorterthan the widely used iteration identification methods such asPSO and GA; that is, we can get the results just after one timeiteration calculation.

The step-by-step implementation procedure can be givenas follows:

(1) initialize the structural parameters of X𝑟,𝑘 with thedata in Table 1 and set iteration variable as 𝑘 = 0;

(2) calculate P𝑘 and J𝑟,𝑘 with (2), (5), X𝑟,𝑘, and the posesacquired from the AACMM;

(3) calculate ΔP𝑘 = P𝑡 − P𝑘 with the reference coordi-nates;

(4) calculate the least squares solution of ΔX𝑟,𝑘 = (J𝑟,𝑘𝑇 ⋅J𝑟,𝑘)−1J𝑟,𝑘ΔP𝑘;

(5) set 𝑘 = 𝑘 + 1 and correct the parameters as X𝑟,𝑘 =X𝑟,𝑘−1 + ΔX𝑟,𝑘 by using the composition ΔX𝑟,𝑘.

4. Experiments

To show the effectiveness of the suggested modeling andidentification algorithms, an experimental study was carriedout based on anAACMM test rig, which is shown in Figure 5.The nominal point repeatability and volumetric accuracyof the AACMM are 0.05mm and 0.06mm, respectively.However, the structural parameters and working principleof the AACMM are not clear for the users. This is themotivation for our current work, which is dedicated toestimate the parameters to achieve an identification model


Figure 5: The articulated arm coordinate measuring machine.

Table 2: The movement uncertainties of AACMM before compensation.

Directions Maximum (max) (mm) Standard deviation (SD) (mm) Average (ave) (mm) Absolute average (AA) (mm)𝑥 1.485 0.639 −0.071 0.524𝑦 1.366 0.478 0.439 0.545𝑧 −0.978 0.358 −0.308 0.421

of the AACMM. We conducted our experiments on thenormal room temperature (in the range of 17∘C to 23∘C)by a same operator. On the other hand, it should be notedthat the main aim of this paper is to provide a parameteridentification method for the AACMM and other similarlink mechanisms. Thus, this paper does not consider thetemperature compensation and calibration test according toASME B.89.4.22, VDI/VDE 2617 Part 9 or ISO 10360 Part 12.

In the data acquisition procedure of the AACMM, manyposes are used to obtain enough information about theAACMM. The information is very important for the follow-ing identification. And acquiring poses is commonly used inthe calibration of AACMM and robots [15–17]. In this paper,we not only obtain many poses but also get the informationof the structural parameters except measuring them directly.We operate the AACMM robot in extensive scenarios, andthe joint angles and reference coordinates of 200 poseswere acquired to test the uncertainty of the AACMM. Thedifferences between the coordinates calculated joint anglesbased on the kinematic model and the reference coordinatesare the movement uncertainty of the AACMM, which isshown in Figure 6. The maximum movement uncertainty is1.485mm, 1.366mm, and −0.978mm in the directions of 𝑥,𝑦,and 𝑧, respectively, and the average of the absolute movementuncertainties are 0.524mm, 0.545mm, and 0.421mm for𝑥, 𝑦, and 𝑧, respectively, as shown in Table 2. Althoughthe values of the structural parameters were estimated afterrepeated tests and measurements, the uncertainties of theAACMM were still relatively large. According to the analysisin Section 3, it is known that suchmovement uncertaintymaybe caused by the uncertainties of the structural parameters.

The significant uncertainties shown in Table 2 are unac-ceptable in practical application as this may severely deteri-orate the overall measurement accuracy. Thus, the identifi-cation of uncertain structural parameters should be furtherconducted, which are then used to compensate the modelingerrors in theDHmodel and then to improve the performance.

From the experimental studies, it is found that themovement uncertainty becomes relatively stable when thenumber of identification poses is greater than 30, and it isalmost invariant when the number of identification poses islarger than 50.Therefore, the joint angles and reference coor-dinates of 50 poses were acquired to identify the structuralparameters of the AACMM, and the identification algorithmpresented in Section 3.3 is used. The identification results areshown in Table 3. And the results can be reached just afterone time iteration. Further experiments show that there is nofurther benefit to increase iteration times.

Compared to Table 1, one may find that the identifiedparameters in Table 3 are very close to the nominal values.This result indirectly validates the efficacy of the suggestedidentification algorithm. There are also some structuralparameters (e.g., 𝑑1, 𝑑6, Δ𝜃6, and 𝛼6) which are not changedfor the reason that they are redundant and do not needto be identified. To further verify the identification results,another 200 groups of data were acquired to test the move-ment uncertainty of the AACMM. Figure 7 shows that themaximummovement uncertainties are −0.112mm, 0.116mm,and 0.117mm in the directions of 𝑥, 𝑦, and 𝑧, respectively,and the averages of the absolute movement uncertainties are0.036mm, 0.038mm, and 0.044mm, which are summarizedin Table 4. It is found from Tables 2 and 4 that the overallaccuracy can be significantly improved by using the proposedidentification and compensation method. The comparisonsbetween the movement uncertainties after identification andthe ones before identification are shown in Figure 8. As itis shown, all the movement uncertainty indexes are greatlyimproved after identification.

5. Conclusions

A constructive parameter identification approach for articu-lated arm coordinatemeasuringmachines has been presentedin this paper. A structural kinematic model is established


Table 3: The identified structural parameters of AACMM.

Linkage number 𝑖 𝑎𝑖 [mm] 𝑑𝑖 [mm] Δ𝜃𝑖 [∘] 𝛼𝑖 [∘]1 −0.121 376.500 0.087 −89.9862 61.942 0.016 0.066 −90.0263 0.0324 750.658 0.001 −89.9974 62.225 −0.795 −0.021 −89.9315 −0.036 500.287 0.071 −89.9566 0.057 15 0 90𝑙 = 97.866mm.

Table 4: The movement uncertainty of AACMM after identification.

Directions Maximum (max) (mm) Standard deviation (SD) (mm) Average (ave) (mm) Absolute average (AA) (mm)𝑥 −0.112 0.044 −0.003 0.036𝑦 0.116 0.046 0.013 0.038𝑧 0.117 0.053 0.007 0.044

x

y

z

20 40 60 80 100 120 140 160 180 2000Number of the poses

−1.5

−1

−0.5

0

0.5

1

1.5

Mov

emen

t unc

erta

inty

(mm

)

Figure 6: The movement uncertainties in the directions of 𝑥, 𝑦, and 𝑧 of AACMM without structural parameter identification andcompensation.

20 40 60 80 100 120 140 160 180 2000Number of the poses

−0.1

−0.05

0

0.05

0.1

0.15

Mov

emen

t unc

erta

inty

(mm

)

x

y

z

Figure 7:Themovement uncertainties in the directions of 𝑥,𝑦, and 𝑧 of AACMMwith structural parameter identification and compensation.


Before identificationAfter identification

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

AAAveSDMax

Mov

emen

t unc

erta

inty

(mm

)

(a) In the direction of 𝑥


AAAveSDMax0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Mov

emen

t unc

erta

inty

(mm

)

(b) In the direction of 𝑦


AAAveSDMax

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Mov

emen

t unc

erta

inty

(mm

)

(c) In the direction of 𝑧

Figure 8: The comparison of the movement uncertainties in the directions of 𝑥, 𝑦, and 𝑧 of AACMM before and after identification.

based on the DH method and verified through experiments.Based on the difference between the coordinates of theprobe calculated by the kinematic model and the referencecoordinates, a mathematical parameter identification modelis further developed to decrease the uncertainties in the DHmodel. The analysis of the Jacobian matrix in the identifica-tion model shows that there are two structural parameters

which are related to others in our case studies. Therefore,these structural parameters are removed from the identifica-tion model.Then the structural parameter identification withthe aim to get the least-square solution of the identificationmodel can be carried out by using the obtained posesand the reference coordinates of the AACMM. To facilitatepractical implementations, experimental studies have been


conducted. These experimental results have revealed theeffectiveness of the proposed structural parameter identifica-tion for AACMM.

The proposed modeling and identification method canbe extended to the calibration of serial robots, where theidentification method is required due to their dynamicoperation environments. This will be further studied in ourfuture work.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no. 51465027).

References

[1] Y. Li and P. R. Nomula, “Surface-opening feature measurementusing coordinate-measuringmachines,” International Journal ofAdvancedManufacturing Technology, vol. 79, no. 9–12, pp. 1915–1929, 2015.

[2] W. Wang, G. Gao, Y. Wu, and Z. Chen, “Analysis and compen-sation of installation errors for circular grating angle sensors,”Advanced Science Letters, vol. 4, no. 6-7, pp. 2446–2451, 2011.

[3] X. H. Li, B. Chen, and Z. R. Qiu, “The calibration anderror compensation techniques for an Articulated Arm CMMwith two parallel rotational axes,” Measurement: Journal of theInternational Measurement Confederation, vol. 46, no. 1, pp.603–609, 2013.

[4] D. Zheng, Z. Xiao, and X. Xia, “Multiple measurement modelsof articulated arm coordinate measuring machines,” ChineseJournal of Mechanical Engineering (English Edition), vol. 28, no.5, pp. 994–998, 2015.

[5] F. Romdhani, F. Hennebelle, M. Ge, P. Juillion, R. Coquet,and J. F. Fontaine, “Methodology for the assessment of mea-suring uncertainties of articulated arm coordinate measuringmachines,”Measurement Science and Technology, vol. 25, no. 12,Article ID 125008, 2014.

[6] A. Piratelli-Filho, F. H. T. Fernandes, and R. V. Arencibia,“Application of virtual spheres plate for AACMMs evaluation,”Precision Engineering, vol. 36, no. 2, pp. 349–355, 2012.

[7] J. Sładek, K. Ostrowska, and A. Gaęska, “Modeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological model,”Measurement, vol. 46, no. 1, pp. 667–679,2013.

[8] Y. Meng and H. Zhuang, “Autonomous robot calibration usingvision technology,”Robotics and Computer-Integrated Manufac-turing, vol. 23, no. 4, pp. 436–446, 2007.

[9] J. U. Dolinsky, I. D. Jenkinson, and G. J. Colquhoun, “Appli-cation of genetic programming to the calibration of industrialrobots,” Computers in Industry, vol. 58, no. 3, pp. 255–264, 2007.

[10] I. Kovac and A. Frank, “Testing and calibration of coordinatemeasuring arms,” Precision Engineering, vol. 25, no. 2, pp. 90–99, 2001.

[11] J. Santolaria, J.-J. Aguilar, D. Guillomıa, and C. Cajal, “Acrenellated-target-based calibration method for laser triangu-lation sensors integration in articulated measurement arms,”

Robotics and Computer-Integrated Manufacturing, vol. 27, no. 2,pp. 282–291, 2011.

[12] J. Santolaria, A. C. Majarena, D. Samper, A. Brau, and J.Velazquez, “Articulated arm coordinate measuring machinecalibration by laser trackermultilateration,”The ScientificWorldJournal, vol. 2014, Article ID 681853, 11 pages, 2014.

[13] H.Hamana,M. Tominaga,M.Ozaki, and R. Furutani, “Calibra-tion of articulated arm coordinate measuring machine consid-ering measuring posture,” International Journal of AutomationTechnology, vol. 5, no. 2, pp. 109–114, 2011.

[14] G. Gao, J. Lu, and H. Yang, “Study on the structrual parametercoupling of articulated arm coordinate measuring machines,”Telkomnika, vol. 11, no. 5, pp. 2454–2459, 2013.

[15] L. Zhu,W. Li, Z. Pan, Y. Guo, andQ.Chen, “Research on param-eter self-calibration method for partly-bonded articulated armcoordinate measuring machine,” Chinese Journal of ScientificInstrument, vol. 35, no. 3, pp. 572–579, 2014.

[16] D. Zheng, Z. Xiao, and Y. Zhou, “Key technologies to improvearticulated arm coordinate measuring machines’ measurementaccuracy,” Journal of Hebei University of Science and Technology,vol. 35, no. 1, pp. 20–23, 2014.

[17] W. Cheng, Y. Fei, L. Yu, and R. Yang, “Probe parameterscalibration for articulated arm coordinatemeasuring machine,”in Proceedings of the 6th International Symposium on PrecisionEngineering Measurements and Instrumentation, vol. 7544 ofProceedings of SPIE, Hangzhou, China, August 2010.

[18] J. Denavit and R. S. Hartenberg, “A kinematic notation forlower-pair mechanisms based on matrices,” ASME Journal ofApplied Mechnics, vol. 22, no. 6, pp. 215–221, 1955.

[19] L. Q. Zhu, W. X. Li, Z. K. Pan, Y. K. Guo, and Q. S. Chen,“Coordinate measuring method with two operation modesbased on the adjustable articulated arms,” Optical Engineering,vol. 53, no. 12, Article ID 122407, 2014.

[20] A. Brau, M. Valenzuela, J. Santolaria, and J. J. Aguilar, “Eval-uation of different probing systems used in articulated armcoordinate measuring machines,” Metrology and MeasurementSystems, vol. 21, no. 2, pp. 233–246, 2014.

[21] R. J. Lapeer, S. J. Jeffrey, J. T. Dao et al., “Using a passivecoordinate measurement arm for motion tracking of a rigidendoscope for augmented-reality image-guided surgery,” Inter-national Journal of Medical Robotics and Computer AssistedSurgery, vol. 10, no. 1, pp. 65–77, 2014.

[22] G.-M. Daniel, B. Joaquın, C. Eduardo, and M.-P. Susana,“Influence of human factor in the AACMM performance: anew evaluationmethodology,” International Journal of PrecisionEngineering and Manufacturing, vol. 15, no. 7, pp. 1283–1291,2014.

[23] G. Gao, W. Wang, and J. Zhou, “Study on the error transferof articulated arm coordinate measuring machines,” TELKOM-NIKA Indonesian Journal of Electrical Engineering, vol. 11, no. 2,pp. 637–641, 2013.

[24] S. Sagara and Y. Taira, “Cooperative manipulation of a floatingobject by some space robots: application of a tracking controlmethodusing the transpose of the generalized Jacobianmatrix,”Artificial Life and Robotics, vol. 12, no. 1-2, pp. 138–141, 2008.

[25] G. Pond and J. A. Carretero, “Formulating Jacobianmatrices forthe dexterity analysis of parallel manipulators,”Mechanism andMachine Theory, vol. 41, no. 12, pp. 1505–1519, 2006.

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